Learning transitions of topological surface codes

Abstract

For the surface code, topological quantum order allows one to encode logical quantum information in a robust, long-range entangled many-body quantum state. However, if an observer probes this quantum state by performing measurements on the underlying qubits, thereby collecting an ensemble of highly correlated classical snapshots, two closely related questions arise: (i) do measurements decohere the topological order of the quantum state; and (ii) how much of the logical information can one learn from the snapshots? Here we address these questions for measurements in a uniform basis on all qubits. We find that for generic measurement angles, sufficiently far away from the Clifford X, Y, and Z directions (such as the X+Y+Z basis) the logical information is never lost in one of the following two ways: (i) for weak measurement, the topological order is absolutely robust; (ii) for projective measurement, the quantum state inevitably collapses, but the logical quantum information is faithfully transferred from the quantum system to the observer in the form of a tomographically complete classical shadow. At these generic measurement angles and in the projective-measurement limit, the measurement ensemble enforced by Born probabilities can be represented by a 2D tensor network that can be fermionized into a disordered, free-fermion network model in symmetry class DIII, which gives rise to a Majorana "metal" phase. When the measurement angle is biased towards the X or Z limits, a critical angle indicates the threshold of a learning transition beyond which the classical shadow no longer reveals full tomographic information (but reduces to a measurement of the logical X or Z state). This learning transition can be described in the language of the network model as a "metal to insulator" transition...

Figures (13)

• Figure 1: (a) Protocol: A surface code is (weakly) measured in the basis $\hat{\sigma}^{\theta,\phi} \equiv \sin\theta \cos\phi X + \sin\theta \sin\phi Y + \cos\theta Z$ for all qubits. The protocol consists of 3 steps: firstly, prepare a surface code in certain state in the logical space; secondly, apply a layer of coherent rotation to all qubits by $e^{i\theta/2 Y}e^{i\phi/2 Z}$; thirdly, apply a layer of weak measurement in the computation basis i.e. Pauli $Z$ basis. The weak measurement is implemented by introducing an ancilla prepared in $|+\rangle$ state ($X\ket{+}=\ket{+}$), and entangling it with the physical qubit by Ising interaction evolution up to time $t\in[0,\pi/4]$, and then measuring it out in the Pauli $Y$ basis. As shown in Ref. NishimoriCat, this ancilla-assisted operation effectively implements a weak measurement gate $e^{\pm \beta Z /2 }$ with tunable measurement strength $\beta=\tanh^{-1}\sin(2t)\in[0,\infty)$. In the following, we will also refer to $t$ as the effective measurement strength. In particular, $t=\pi/4$ describes the projective-measurement limit with infinite strength $\beta=\infty$. (b) Three-dimensional Born average phase diagram of parameter $(t,\theta,\phi)$: the radial coordinate corresponds to the compactified measurement strength $t$, while the direction of the coordinate corresponds to the measurement basis of the physical qubit, akin to the Bloch sphere of a spin-$1/2$, that can be parametrized by the spherical coordinates $(\theta,\phi)$. The blue surface denotes the critical boundary of surface code, which sets the learnability threshold. Below the threshold $t<t_c$, the surface code remains topological, and the measurement outcomes do not extract useful information about the logical state in thermodynamic limit. In the conventional measurement basis such as $X$ or $Z$ direction, a finite threshold $t_c=0.143\pi$Puetz25percolationteleportcodeNishimoriCatWan25nishimori is established by the Nishimori transition (blue dot). The same Nishimori universality describes the whole transition line (blue dots) until the self-dual limit at $X+Z$, which is captured instead by the weak self-dual universality class Wang25selfdual. Around the self-dual directions (purple line), $t_c=\pi/4$ ($\beta_c=\infty$), the maximal infinite threshold is found to hold for a big phase region (denoted by the purple surface) enveloped by the red doted line as the boundary.
• Figure 2: projective-measurement limit. (a) A deterministic isometry matrix encodes the logical qubit to the physical qubits of the surface code. The logical qubit is initially maximally entangled with the reference ($R$) qubit. The physical qubits are subjected to projective measurement, whose outcome is recorded by an observer ($A$). For each observed measurement record $\mathbf{s}$, the observer ($A$) can decode (by inverting the unitary encoder that prepares the surface code state) the corresponding state of reference ($R$) qubit, which is a snapshot of the logical qubit. These measurement records $\mathbf{s}$ form the raw data from which we construct classical shadows of the logical qubit state for tomography HuangKuengPreskill2020shadowtomography. (b) Taking together all the possible measurement records, $\{P(\mathbf{s}), \ket{\psi_R(\mathbf{s})}\}$ forms a projected ensemble of $R$. In the gapped phases (white), the ensemble converges to a bimodal distribution, revealing the logical $Z$ or logical $X$ state. At the self-dual critical point $X+Z$, the ensemble delocalizes to a uniform distribution, but confined to the real $XZ$ plane, forming a ring on the equator of the Bloch sphere. In the metal gapless phase (purple), the ensemble fully delocalizes to a Haar measure uniform coverage of the Bloch sphere. Nevertheless, it crossovers to the bimodal distribution at the Clifford $Y$ limit, which is a singular point. The shown ensembles of states are computed for surface code of code distance $d=8$. (c) The boundary entanglement entropy has distinct scaling at the phases. In the gapped phases, it exhibits area law. At self-dual critical point, critical logarithmic scaling is found. Into the metal gapless phase, $\log^2 L$ dominates the entanglement entropy scaling. The $Y$ limit corresponds to a ballistic metal, exhibiting volume law scaling. The Y-axis is described by purely unitary evolution. By contrast, the X and Z axes are described by pure measurement evolutions. The circuit and the boundary state is shown in Fig. \ref{['fig:bilayertensornetwork']}(c).
• Figure 3: Random 2D tensor network and (1+1)D quantum circuit. The top and the bottom layer of the tensor network captures the bra and ket of the quantum state as a PEPS. (a) For weak measurement, the Kraus operator $M_{\mathbf{s}}^\dagger M_\mathbf{s}$ glues the two layers. (b) For projective measurement, $M^\dagger_\mathbf{s} M_\mathbf{s}$ is a projector, such that $P(\mathbf{s}) = |\bra{\mathbf{s}}\ket{\psi}|^2$, which means the two layers can be factorized. As will be shown in Sec. \ref{['subsec:transfermatrix']}, each layer alone corresponds to a random Ising model with complex coupling, which can be fermionized into a free fermion network model. In this aspect, weakening the measurement strength in (a) induces fermion interactions. Both (a) and (b) as 2D tensor network can be interpreted as (1+1)D (non-unitary) quantum circuit that propagates a 1D quantum chain in the space direction, updated by the transfer matrix. (c) The ket layer of tensor network in (b) can be identified as a monitored (1+1)D quantum circuit. The gate parameters are controlled by the coherent rotation $U(\theta, \phi)$ and the measurement outcome $\mathbf{s}$. Here we show only the ket layer of the state. The gates are derived from Eq. \ref{['eqn:ZZgate']}\ref{['eqn:Xgate']}.
• Figure 4: Learning transitions by tuning the measurement strength. Shown are two cuts in the $XY$ plane for finite measurement strength $t$, one far from and one close to the $Y$-point, shown on the left/right respectively. The top row shows the coherent information, while the bottom row shows the boundary entanglement entropy. (a) For a cut at an angle $\phi=0.1\pi$ (close to the $X$ point in the $XY$ plane) the transition occurs at a finite measurement strength with the coherent information exhibiting a crossing point at a critical $t_c$. The boundary entanglement entropy exhibits a peak that approaches the transition point for larger system sizes with $\log(L)$ scaling, characteristic of the scaling at a (1+1)D quantum critical point. In the projective-measurement limit of $t=\pi/4$ the entanglement entropy decreases with system size, indicating an area law phase. (b) For a cut at an angle $\phi=0.4\pi$ (close to the $Y$ point in the $XY$ plane) the transition occurs only in the projective measurement limit ($t=\pi/4$), where the coherent information exhibits a collapse. The boundary entanglement entropy, in contrast, exhibits an entanglement plateau for finite system sizes, whose height scales linearly with code distance, reminiscent of a volume-law scaling.
• Figure 5: Projected ensemble of the reference qubit on (a) the XZ line, (b) the self-dual line, and (c) the XY line. The spheres above each plot show the projected ensemble distribution for different lines on the Bloch sphere labeled by $\alpha$ in the phase diagram by displaying $10^4$ sampled points from the distribution. The $X$ and $Z$ points on the Bloch sphere of highly concentrated densities are marked with small red spheres. In the above plots, the Kullback-Leibler divergence (relative entropy) between the projected ensemble and a uniform (Haar) distribution of the Bloch sphere is normalized by the bimodal distribution value $\tilde{D}(P||Q) = D(P||Q) (\log n - \log 2)$ where $n$ is the number of discrete patches covering the sphere. For the ring distribution, $\tilde{D}(P||Q)$ becomes $1 - \log(m) / \log(n)\to 1/2$, where $m$ is the number of patches covering the equator as a 1D line, such that the ratio of dimension is $\log m / \log n \to 1/2$ for the limit of high resolutions. In the $n \to \infty$ limit, the dip in (a) at $\theta = \pi/4$ and the first data point at $X+Z$ in (b) reach exactly $\tilde{D}(P||Q) = 1/2$. See Appendix \ref{['app:KL_divergence']} for more details.